4x√(2x^2 + 2x - 1). find first derivative
To find the first derivative, you can use the product rule. The product rule states that if you have two functions u(x) and v(x) that are both differentiable, the derivative of their product u(x) * v(x) is given by:
(d(uv)/(dx)) = u * (dv/dx) + v * (du/dx)
In this case, u(x) = 4x and v(x) = √(2x^2 + 2x - 1). Let's find the derivatives of u(x) and v(x) first.
The derivative of u(x) = 4x with respect to x is found by applying the power rule, which states that d(x^n)/dx = n * x^(n-1), where n is a constant.
So, the derivative of u(x) = 4x is du/dx = 4.
To find the derivative of v(x) = √(2x^2 + 2x - 1), we need to use the chain rule. Let's rewrite v(x) as (2x^2 + 2x - 1)^(1/2) to make it easier to differentiate.
The derivative of v(x) = (2x^2 + 2x - 1)^(1/2) is given by:
(dv/dx) = (1/2) * (2x^2 + 2x - 1)^(-1/2) * (d(2x^2 + 2x - 1)/dx)
Now, let's find the derivative of 2x^2 + 2x - 1.
The derivative of 2x^2 + 2x - 1 with respect to x can be found by applying the power rule to each term. The power rule states that the derivative of x^n with respect to x is n * x^(n-1).
So, the derivative of 2x^2 is d(2x^2)/dx = 4x, the derivative of 2x is d(2x)/dx = 2, and the derivative of -1 is 0.
Therefore, the derivative of 2x^2 + 2x - 1 with respect to x is d(2x^2 + 2x - 1)/dx = 4x + 2.
Now, substituting the values we found, we have:
(dv/dx) = (1/2) * (2x^2 + 2x - 1)^(-1/2) * (4x + 2)
Finally, applying the product rule, we can find the first derivative of 4x√(2x^2 + 2x - 1) as:
(d(4x√(2x^2 + 2x - 1))/dx) = u * (dv/dx) + v * (du/dx) = 4x * [(1/2) * (2x^2 + 2x - 1)^(-1/2) * (4x + 2)] + √(2x^2 + 2x - 1) * 4